Classical control theory is concerned with improving a controlled system's performance measures in both the frequency and time domains. In particular, improvement objectives of time-domain performances generally involve decreasing rise-times, steady state error, sensitivity to plant uncertainty or external disturbances, and settling-time responses of a given linear time-invariant system. Likewise, improvement objectives in frequency-domain performances generally involve increasing phase and gain stability margins to improve the stability of a given linear time-invariant system. The introduction of a feedback controller, or compensator, to a control system loop is a manner by which to achieve these improvements.
Prior art feedback compensators employing various designs to improve system performances are numerous. The simplest form of compensation used to improve the transient response of a system is based on high gain feedback, as it is well known that increasing gain beneficially results in increased response speeds, decreased steady state error, and the like. However, high gain compensation requires a compromise between the selection of a proper gain and other acceptable performance measures. Indeed, a gain increase to a high enough extent in certain systems can lead to oscillatory behavior and instability.
In practice, the most widely used industrial compensator is a Proportional-Integral-Derivative (PID) and tuning of PID controllers to meet performance specifications is based on varying approaches. Prior art frequency response tuning techniques based on the theories of Nyquist, Bode, Evans and others are generally known to facilitate such tuning. Similarly, many time-domain tuning approaches are also provided for in the prior art. One particular approach to feedback controller tuning is the pole placement or pole assignment design method. This method entails identifying desirable poles based on the understanding of how the location of the poles in the complex S-domain influences the transient response of a controlled system and subsequently determining the feedback gain, for example the state proportional term of a PID controller, so that the closed control loop displays these required poles.
One drawback, however, is that each prior art tuning approach is optimal with respect to a selected measure of performance and a compromise between desired behavior and technical limitations must be made. For instance, the pole placement tuning method is suitable for tuning transient response performance yet is not adept at enhancing other common design specifications such as disturbance rejection, noise sensitivity and stability margins. Moreover, a PID compensator cannot secure any phase margin when the gain increases unboundedly, causing instability and oscillation, should a plant have more than three poles in excess of its zeroes. Furthermore, the addition of lead-lag compensation to speed up transient response and improve steady state response increases controller complexity when the gain is increased thus requiring a design trade-off between bandwidth performance and compensator complexity.
As a result of these shortcomings, quasi-linear compensators have been proposed. Quasi-linear compensators eliminate the contradiction between performance and compensator complexity and consequentially achieve arbitrary close to perfect tracking performance when the gain of the compensator tends to infinity (see KELEMEN Mattei, BENSOUSSAN DAVID, “On the Design, Robustness, Implementation and Use of Quasi-Linear Feedback Compensator”, International Journal of Control, 15 Apr. 2004, Vol 77, No 6, pp 527-545) which is incorporated herein by reference. Furthermore, quasi-linear feedback compensators have been shown to have non-oscillatory time responses for high compensator gains. These benefits which quasi-linear compensators provide over linear compensators are explained by the automatic adaptation of the closed loops poles to stability and stability margins for higher system gains. However, prior art quasi-linear controllers have yet to comprehensively address all performance considerations, in particular the improvement of system rise times.
What is therefore needed, and an object of the present invention, is a quasi-linear controller that is simultaneously optimisable in the time-domain and the frequency domain, which achieves arbitrarily fast and robust tracking, improved gain and phase stability margins, improved time domain performances, and improved sensitivity of a variety of stable and unstable systems.
What is also needed is a quasi-linear compensator that applies to a large family of invertible systems that are stable and unstable and have any number of poles in excess of zeros.